We are given the expression:

$\frac{\cot^2(x) \cos(x)}{\csc(x) - 1}$

Step 1: Rewrite in terms of sine and cosine

First, express $\cot(x)$ and $\csc(x)$ in terms of sine and cosine:

$\cot(x) = \frac{\cos(x)}{\sin(x)} \quad \text{and} \quad \csc(x) = \frac{1}{\sin(x)}$

Now substitute these into the original expression:

$\frac{\left( \frac{\cos^2(x)}{\sin^2(x)} \right) \cos(x)}{\frac{1}{\sin(x)} - 1}$

Step 2: Simplify the numerator and denominator

First, simplify the numerator:

$\frac{\cos^2(x)}{\sin^2(x)} \cdot \cos(x) = \frac{\cos^3(x)}{\sin^2(x)}$

Now simplify the denominator:

$\frac{1}{\sin(x)} - 1 = \frac{1 - \sin(x)}{\sin(x)}$

Step 3: Combine the fraction

The expression now looks like this:

$\frac{\frac{\cos^3(x)}{\sin^2(x)}}{\frac{1 - \sin(x)}{\sin(x)}}$

This is a complex fraction. To simplify it, multiply the numerator by the reciprocal of the denominator:

$\frac{\cos^3(x)}{\sin^2(x)} \cdot \frac{\sin(x)}{1 - \sin(x)} = \frac{\cos^3(x) \cdot \sin(x)}{\sin^2(x) \cdot (1 - \sin(x))}$

Step 4: Simplify the result

Cancel one factor of $\sin(x)$ from the numerator and denominator:

$\frac{\cos^3(x)}{\sin(x) \cdot (1 - \sin(x))}$

This is the expression without a fraction.

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Would you like more details on any step or further clarification?

Here are five related questions to expand on this topic:

1. How can we simplify expressions involving both sine and cosine further?
2. How do cotangent and cosecant relate to other trigonometric identities?
3. What strategies can be used to simplify complex fractions?
4. How does this expression behave for specific values of $x$, such as $\pi/2$ or $\pi$?
5. What are some real-world applications of simplifying trigonometric expressions?

Tip: When dealing with trigonometric identities, converting everything into sine and cosine often simplifies the process!